{
 "metadata": {
  "name": "",
  "signature": "sha256:c582638faa951e64175ca28a0ce3d8afc97a0a04ae5ab0bb8e27cab77d9aa65b"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "from __future__ import  division\n",
      "from IPython.html.widgets import interact,fixed\n",
      "from scipy import stats\n",
      "import sympy\n",
      "from sympy import S, init_printing, Sum, summation,expand\n",
      "from sympy import stats as st\n",
      "from sympy.abc import k\n",
      "import re"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Maximum likelihood estimation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\\hat{p} = \\frac{1}{n} \\sum_{i=1}^n X_i$$\n",
      "\n",
      "$$ \\mathbb{E}(\\hat{p}) = p $$\n",
      "\n",
      "$$ \\mathbb{V}(\\hat{p}) = \\frac{p(1-p)}{n} $$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy.plot( k*(1-k),(k,0,1),ylabel='$n^2 \\mathbb{V}$',xlabel='p',fontsize=28)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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9rYiIx778Ei6+2JwV0rCh02kMtUDO4ZlnYNQoFQ8RcV50NPTtC7NnO53kJ2qB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\nrkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x5872bf0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "<sympy.plotting.plot.Plot at 0xcfc25f0>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b=stats.bernoulli(p=.8)\n",
      "samples=b.rvs(100)\n",
      "print var(samples)\n",
      "print mean(samples)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.1659\n",
        "0.79\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def slide_figure(n=100,m=1000,p=0.8):\n",
      "    fig,ax=subplots()\n",
      "    ax.axis(ymax=25)\n",
      "    b=stats.bernoulli(p=p)\n",
      "    v=iter(b.rvs,2)\n",
      "    samples=b.rvs(n*1000)\n",
      "    tmp=(samples.reshape(n,-1).mean(axis=0))\n",
      "    ax.hist(tmp,normed=1)\n",
      "    ax1=ax.twinx()\n",
      "    ax1.axis(ymax=25)\n",
      "    ax1.plot(linspace(0,1,200),stats.norm(mean(tmp),std(tmp)).pdf(linspace(0.0,1,200)),lw=3,color='r')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "interact(slide_figure, n=(100,500),p=(0.01,1,.05),m=fixed(500));"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0xd1cd570>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Maximum A-Posteriori (MAP) Estimation"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.abc import p,n,k\n",
      "\n",
      "sympy.plot(st.density(st.Beta('p',3,3))(p),(p,0,1) ,xlabel='p')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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NA4YPB375BVi7FqhRQ3UiYdjCnpgoPZO//hpo08Zmn5aIqEKKi2UZZGGhHLfp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\nZDIs7EREJsPCTkRkMizsREQm8/+piiqlCc+7MwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0xd6e0e10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "<sympy.plotting.plot.Plot at 0x58725f0>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Maximize the MAP function to get form of estimator"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "obj=sympy.expand_log(sympy.log(p**k*(1-p)**(n-k) * st.density(st.Beta('p',6,6))(p)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sol=sympy.solve(sympy.simplify(sympy.diff(obj,p)),p)[0]\n",
      "print sol"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "(k + 5)/(n + 10)\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$ \\hat{p}_{MAP} = \\frac{(5+\\sum_{i=1}^n X_i )}{(n + 10)}  $\n",
      "\n",
      "with corresponding expectation \n",
      "\n",
      "$ \\mathbb{E}  = \\frac{(5+n p )}{(n + 10)} $\n",
      "\n",
      "which is a biased estimator. The variance of this estimator is the following:\n",
      "\n",
      "$$ \\mathbb{V}(\\hat{p}_{MAP}) = \\frac{n (1-p) p}{(n+10)^2} $$\n",
      "\n",
      "compare this to the variance of the maximum likelihood estimator, which is reproduced here:\n",
      "\n",
      "$$ \\mathbb{V}(\\hat{p}_{ML}) = \\frac{p(1-p)}{n} $$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n=5\n",
      "def show_bias(n=30):\n",
      "    sympy.plot(p,(5+n*p)/(n+10),(p,0,1),aspect_ratio=1,title='more samples reduce bias')\n",
      "    \n",
      "interact(show_bias,n=(10,500,10));"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0xd8457f0>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Compute the variance of the MAP estimator"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sum(sympy.var('x1:10'))\n",
      "expr=((5+(sum(sympy.var('x1:10'))))/(n+10))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def apply_exp(expr):\n",
      "    tmp=re.sub('x[\\d]+\\*\\*2','p',str(expand(expr)))\n",
      "    tmp=re.sub('x[\\d]+\\*x[\\d]+','p**2',tmp)\n",
      "    tmp=re.sub('x[\\d]+','p',tmp)\n",
      "    return sympy.sympify(tmp)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ex2 = apply_exp(expr**2)\n",
      "print ex2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "8*p**2/25 + 11*p/25 + 1/9\n"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tmp=sympy.simplify(ex2 - (apply_exp(expr))**2 )\n",
      "sympy.plot(tmp,p*(1-p)/10,(p,0,1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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cSMnVzk52hOYlM5PGcv78k2aImSpOBhWYMYNmjHz+udQwzIIQdNe6bRs9mjWj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\nyYAxxhg4GTDGGAPw/1VjRDbaCH6TAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xd6d7810>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "<sympy.plotting.plot.Plot at 0xd6d7410>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "General case"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def generate_expr(num_samples=10,alpha=6):\n",
      "    n = sympy.symbols('n')\n",
      "    obj=sympy.expand_log(sympy.log(p**k*(1-p)**(n-k) * st.density(st.Beta('p',alpha,alpha))(p)))\n",
      "    sol=sympy.solve(sympy.simplify(sympy.diff(obj,p)),p)[0]\n",
      "    expr=sol.replace(k,(sum(sympy.var('x1:%d'%(num_samples)))))\n",
      "    expr=expr.subs(n,num_samples)\n",
      "    ex2 = apply_exp(expr**2)\n",
      "    ex =  apply_exp(expr)\n",
      "    return (ex,sympy.simplify(ex2-ex**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "num_samples=10\n",
      "X_bias,X_v = generate_expr(num_samples,alpha=2)\n",
      "p1=sympy.plot(X_v,(p,0,1),show=False,line_color='b',ylim=(0,.03),xlabel='p')\n",
      "X_bias,X_v = generate_expr(num_samples,alpha=6)\n",
      "p2=sympy.plot(X_v,(p,0,1),show=False,line_color='r',xlabel='p')\n",
      "p3=sympy.plot(p*(1-p)/num_samples,(p,0,1),show=False,line_color='g',xlabel='p')\n",
      "p1.append(p2[0])\n",
      "p1.append(p3[0])\n",
      "p1.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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zbo6/CcPq4td4UASpqcCMGXwlseXL+Zw1SpK/iIIoLORjFA4c4OtNL10KKMMi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\nTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xd57afb0>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p1=sympy.plot(n*(1-p)*p/(n+10)**2,(p,0,1),show=False,line_color='b',ylim=(0,.05),xlabel='p',ylabel='variance')\n",
      "p2=sympy.plot((1-p)*p/n,(p,0,1),show=False,line_color='r',ylim=(0,.05),xlabel='p')\n",
      "p1.append(p2[0])\n",
      "p1.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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aauNKb2tZuBCIjwfWrdM7EiKyJxs3SsXqAwcAC8d17RVXeltDcbGMXbz0kt6R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\nAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xd5b1ed0>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def show_variance(n=5):\n",
      "    p1=sympy.plot(n*(1-p)*p/(n+10)**2,(p,0,1),show=False,line_color='b',ylim=(0,.05),xlabel='p',ylabel='variance')\n",
      "    p2=sympy.plot((1-p)*p/n,(p,0,1),show=False,line_color='r',ylim=(0,.05),xlabel='p')\n",
      "    p1.append(p2[0])\n",
      "    p1.show()    \n",
      "interact(show_variance,n=(10,120,2));\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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kZPURFQV06eJy+w7VnWnTZE27desM6TTy3MCYORNYsEBmVpponLMebDZ5bLRr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\nGEREpAkDg4iINPl/idlgn/UiREQAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xd6f4e70>"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The obvious question is what is the value of  a biased estimator? The key fact is that the MAP estimator is biased, yes, but it is biased   according to the prior probability of $\\theta$. Suppose that the true parameter $p=1/2$ which is exactly at the peak of the prior probability function. In this case, what is the bias?\n",
      "\n",
      "$ \\mathbb{E}  = \\frac{(5+n p )}{(n + 10)} -p \\rightarrow 0$\n",
      "\n",
      "and the variance of the MAP estimator at this point is the following:\n",
      "\n",
      "$$ \\frac{n p (1-p)}{(n+10)^2} \\rightarrow $$\n",
      "\n",
      "\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pv=.60\n",
      "nsamp=30\n",
      "fig,ax=subplots()\n",
      "rv = stats.bernoulli(pv)\n",
      "map_est=(rv.rvs((nsamp,1000)).sum(axis=0)+5)/(nsamp+10);\n",
      "ml_est=(rv.rvs((nsamp,1000)).sum(axis=0))/(nsamp);\n",
      "_,bins,_=ax.hist(map_est,bins=20,alpha=.3,normed=True,label='MAP');\n",
      "ax.hist(ml_est,bins=20,alpha=.3,normed=True,label='ML');\n",
      "ax.vlines(map_est.mean(),0,12,lw=3,linestyle=':',color='b')\n",
      "ax.vlines(pv,0,12,lw=3,color='r',linestyle=':')\n",
      "ax.vlines(ml_est.mean(),0,12,lw=3,color='g',linestyle=':')\n",
      "ax.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 23,
       "text": [
        "<matplotlib.legend.Legend at 0xe198e70>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0xd9226d0>"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    }
   ],
   "metadata": {}
  }
 ]
}